WARM - UP March 8- Tuesday Find the geometric mean of each pair of numbers: 1. ½ and 1 2. 2 and 8 3. 1 and 1000 =𝟏 =𝟐 =𝟏𝟎 𝟏𝟎 4. a) Write a similarity statement relating the three triangles in the diagram. PQR  PSQ   QSR b) If PS = 21 and SR = 4, find the lengths of 𝑃𝑄 , 𝑄𝑆 , and 𝑄𝑅 . 𝑷𝑸=𝟓 𝟐𝟏 𝑸𝑺=𝟐 𝟐𝟏 𝑸𝑹=𝟏𝟎

9-4 Similarity in Right Triangles Right Triangle Similarity Theorem The altitude to the hypotenuse of a right  divides the  into two s that are similar to the original  and to each other. 9-4 Similarity in Right Triangles Corollary 1: 9-4 Similarity in Right s Corollary 2: Geometric Mean (Leg) Theorem Geometric Mean (Altitude) Theorem The length of the altitude to the hypotenuse of a right  is the geometric mean of the lengths of the segments of the hypotenuse. The altitude to the hypotenuse of a right  separates the hypotenuse so that the length of each leg of the  is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to the leg. 

9-4 Similarity in Right Triangles Real – World Problem Applications PROBLEM #1 (Civil Engineering) Study the plan below. A service station will be built on the highway, and a road will connect it with Cray. How far from Blare should the service station be located so that the proposed road will be perpendicular to the highway? b. How long will the new road be? Let y = length of the new road Solution: 𝟑𝟐 𝒚 = 𝒚 𝟏𝟖 Let x = distance of the Service Station from Blare 𝒚 𝟐 =𝟑𝟐 𝟏𝟖 𝒙+𝟑𝟐 30 = 30 𝒙 𝒙−𝟏𝟖 𝒙+𝟓𝟎 =𝟎 𝒚= 𝟑𝟐 𝟏𝟖 𝒙−𝟏𝟖 =𝟎 𝒙+𝟓𝟎 =𝟎 𝒙 𝒙+𝟑𝟐 = 𝟑𝟎 𝟐 𝒚=𝟐𝟒 𝒙=𝟏𝟖 𝒙=−𝟓𝟎 𝒙 𝟐 +𝟑𝟐𝒙=𝟗𝟎𝟎 𝐚𝐧𝐬𝐰𝐞𝐫:𝐛) 𝒚=𝟐𝟒 𝒎𝒊 𝐚𝐧𝐬𝐰𝐞𝐫:𝐚) 𝒙=𝟏𝟖 𝒎𝒊 𝒙 𝟐 +𝟑𝟐𝒙−𝟗𝟎𝟎=𝟎

9-4 Similarity in Right Triangles Real – World Problem Applications PROBLEM #2 (The Totem Pole) To estimate the height of a totem pole, Jorge uses a small square of plastic. He holds the square up to his eyes and walks backward from the pole. He stops when the bottom of the pole lines up with the bottom edge of the square and the top of the pole lines up with the top edge of the square. Jorge’s eye level is about 2 m from the ground. He is about 3 m from the pole. Which is the best estimate for the height of the totem pole?

9-4 Similarity in Right Triangles Real – World Problem Applications PROBLEM #3 (STEM) The architect’s side view drawing of a saltbox-style house shows a post that supports the roof ridge. The support post is 10 ft tall. How far from the front of the house is the support post positioned?

9-4 Similarity in Right Triangles PROBLEM #4 (1F. Analyze Mathematical Relationships) 𝐶𝐷 is the altitude to the hypotenuse of right ABC. The coordinates of A, D, and B are (4, 2), (4, 6), and (4, 15), respectively. Find all possible coordinates of point C.

9-4 Similarity in Right Triangles Exit Ticket The roof of a house forms a right angle, with each side of the roof measuring 28 ft in length. Find the width and height of the roof.